[FOM] interpretations

[Ali Enayat]

In his posting of December 15, Thomas Forster wrote:

>>I have a student writing an essay on the Forti-Honsell Antifoundation
>>Axiom (as in Aczel's delightful book).  I have been telling him that he
>>might like to think about the mutual interpretability of ZF + foundation
>>and ZF + antifoundation.

Note that in this discussion ZF does not include Foundation, aka
Regularity.

Albert Visser and myself have some unpublished results about this
matter that will hopefully see the light of day soon; but I do not
know of any other published work that specifically this issue. However, the
following paper might be of interest to your student (it shows that ZF
+ Foundation + urelements is definitionally equivalent to ZF + Foundation).

Set Theory With and Without Urelements and Categories of
Interpretations, by Benedikt Löwe, in Notre Dame J. of Formal Logic
vol 43, pp.83-91, (2006).

Forster has also written:

>Interpretations coming in varying degrees of
>niceness and i am wondering what would be some good literature to start
>him off on.   I think immediately of Friedman and Visser but there is
>surely other literature out there that my man could read with profit.

In addition to work of Friedman and Visser, I recommend the following:

Aspects of Incompleteness, by Per Lindström, Lecture Notes in Logic (2003)

(the above focuses on arithmetic, but many of the results hold in a
wider context, including ZF).

A Lattice of Chapters of Mathematics: Interpretations Between Theorems ,
by Jan Mycielski, Pavel Pudlák, and Alan S. Stern , Memoirs of AMS, 1990.

Regards,

Ali Enayat